attaching things (eg coat hooks) to MDF trim

Okay, so my mudroom has wainscot all around topped with a flat trim board that is about 6 inches tall, 1/2 inch thick that is MDF.

My builder said that he would install the many coat hooks I have onto this trim board for me as it could be tricky. (???)

Of course he never did it, so I think I'll just get out a drill and do it. Can anyone recommend a certain bit and/or a certain type of screw? Is this stuff going to just crumble? Perhaps I should go all the way through the sheetrock? I really want these hooks to be secure...